How the Web designer can become a UI/UX Designer?

I'd say you need a good base.

1. To start with – design tools. While great designers are efficiently using their weapons of choice, tools themselves never made anybody a great designer. Most of them are so far from the true nature of the design that they can only spoil you.
2. Turn yourself into a constant redesign machine. Think about how to improve any product and service that you see. Including a door handle and a coffee machine.
3. Observe people around you and analyse patterns in their behaviour, problems that they encounter using certain products etc.
4. Whenever you’re tempted to design something, stop yourself and try to describe: who’s the Customer, what’s the Problem and what could solve the Problem (the C-P-S Hypothesis).
5. Design as little as you can. Solve problems.